Minimal entropy and uniqueness of price equilibria in a pure exchange economy
aa r X i v : . [ ec on . T H ] F e b Minimal entropy and uniqueness of priceequilibria in a pure exchange economy ∗ Andrea Loi † Stefano Matta ‡ (University of Cagliari)February 22, 2021 Abstract :We introduce uncertainty into a pure exchange economy and establish a connectionbetween Shannon’s differential entropy and uniqueness of price equilibria. Thefollowing conjecture is proposed under the assumption of a uniform probabilitydistribution: entropy is minimal if and only if the price is unique for every economy.We show the validity of this conjecture for an arbitrary number of goods and twoconsumers and, under certain conditions, for an arbitrary number of consumersand two goods.
Keywords:
Entropy, uniqueness of equilibrium, price multiplicity, equilibriummanifold, minimal submanifold.
JEL Classification:
D50, D51, D80. ∗ The first author was supported by INdAM. GNSAGA - Gruppo Nazionale per le StruttureAlgebriche, Geometriche e le loro Applicazioni, by KASBA Funded by Regione Autonoma dellaSardegna and by STAGE- Funded by Fondazione di Sardegna.The second author was supportedby GROVID and by STAGE, funded by Fondazione di Sardegna and Regione Autonoma dellaSardegna. † Dipartimento di Matematica e Informatica, email: [email protected]. ‡ Correspondence to S. Matta, Dipartimento di Economia, University of Cagliari, viale S. Ig-nazio 17, 09123 Cagliari, Italy, tel. 00390706753340, Fax 0039070660929 email: [email protected] Introduction
In a pure exchange economy let us denote by x = ( p, ω ) an initial allocation ω and its supporting equilibrium price vector p . Suppose that x is slightly perturbedby exogenous, i.e. shocks, or endogenous factors, e.g. the uncertainty related tothe effects of Safra’s competitive manipulation [19]. The result of this perturbationis a new allocation and equilibrium price vector, x ′ , belonging to a neighborhood N of x . We can represent this process as a probability model, where the randomvariable ranges in the set N . Observe that N is not an Euclidean space. It belongsto a space of endowments and prices and consists of points such that aggregateexcess demand function is equal to zero. Under standard smooth assumptionsand in a fixed total resource setting, N becomes a submanifold with boundary ofa manifold called the equilibrium manifold , denoted by E ( r ) which in turn is asmooth submanifold of S × Ω( r ), where S is the space of prices and Ω( r ) the spaceof economies (see the seminal work by Balasko [2] and also Section 2).Thus E ( r ) can be equipped with a natural measure, namely the Riemannianvolume form dM g associated to the Riemannian metric g induced by the flat metricof its ambient space S × Ω( r ) (see e.g. [13]). The probability that x ∈ E ( r ) belongsto N is P r ( x ∈ N ) = Z N p ( x ) dM g ( x ) , where p is a given probability density on E ( r ) (the reader is referred to [18] fora geometric approach to probability theory on Riemannian manifolds). Moreover,following Shannon [20] (see also [6]) in this framework we define the differentialentropy of N as H ( N ) = − Z N p ( x ) log ( p ( x )) dM g ( x ) . Obviously when E ( r ) is an Euclidean flat space then one can take dM g equals tothe Lebesgue measure and H ( N ) is the differential entropy defined in [20].Since entropy is a measure of missing information it is natural to investigateunder which conditions it is minimized. Therefore we provide he following: Definition (MEP)
The equilibrium manifold satisfies the minimal entropy prop-erty (MEP) if for every economy x belonging to E ( r ) , there exists a neighborhood N of x in E ( r ) such that H ( N ) ≤ H ( ˜ N ) , where ˜ N is any other submanifold of S × Ω( r ) containing x which has the same boundary of N and whose volume struc-ture, in the same way as N , is induced by the flat metric of the ambient space S × Ω( r ) . It would be interesting (and challenging) to study how the choice of differentprobability distributions affects the economic properties implied by (MEP). Thisissue, which deserves further analysis, is beyond the scope of this paper.2n the other hand, it is natural to restrict to the case of uniform distribution,namely when the probability density function is given by p N = χ n V ( N ) , where V ( N ) = R N dM g ( x ) is the volume of N and χ N : E ( r ) → { , } is thecharacteristic function supported on N . Under this assumption H ( N ) = − Z N V ( N ) log ( 1 V ( N ) ) dM ( x ) = log ( V ( N )) , (1)and so by the increasing property of the logarithm we deduce that in the caseof uniform distribution the (MEP) is equivalent to the following Definition (MVP)
The equilibrium manifold satisfies the minimal volume prop-erty (MVP) if for every economy x belonging to E ( r ) , there exists a neighborhood N of x in E ( r ) such that V ( N ) ≤ V ( ˜ N ) , where ˜ N is any other submanifold of S × Ω( r ) containing x which has the same boundary of N . Now the (MVP) for E ( r ) can be translated into the language of differentialgeometry: the equilibrium manifold E ( r ) is a stable minimal submanifold of S × Ω( r ), i.e. a local minimum of the volume functional. In particular E ( r ) is a criticalpoint of the volume functional, namely E ( r ) is a minimal submanifold of S × Ω( r ).Observe now that according to Theorem 2.1 below, if for every economy thereis uniqueness of equilibrium, the equilibrium manifold is “flat” (and hence mini-mal): i.e., (global) uniqueness implies (MVP). Here we explore the reverse of thisimplication: if there is price multiplicity, can (MVP) holds true? In other words,does (MVP) implies uniqueness? This is not a trivial issue: in fact the equilibriummanifold can almost arbitrarily be twisted for an appropriate preference profile .Hence one could expect to find an utility profile which gives rise to multiplicityand minimality. Actually, we believe that exactly the opposite is true. Indeed weaddress the following: Conjecture:
Under the assumption of uniform distribution the equilibrium man-ifold satisfies (MEP) if and only if the equilibrium price is unique. Throughout this paper we will content ourselves with this definition since in the proof ofour main results we are not using the differential geometric machinery of the theory of minimalsubmanifolds. The interested reader is referred to [21] for more details and material on minimalsubmanifolds. The simplest examples of minimal submanifolds arise when n = 1: in this casethey are simply geodesics of the ambient space. In higher dimensions every totally geodesicsubmanifold is a minimal submanifold (cf. also [11] for some properties of geodesics and totallygeodesic submanifolds of the equlibrium manifold). Nevertheless, there exist a lot of interestingminimal submanifolds (see [21] or Section 3 below). Even if the equilibrium manifold E ( r ) is unknotted in its ambient space [7].
3n other words, we believe that an utility profile which minimizes entropy (andhence volume) with uniform distribution is incompatible with price multiplicity.In this paper we show the validity of this conjecture in the case of an arbi-trary number of goods and two consumers (Theorem 3.1) and in the case of anarbitrary number of consumers and two goods (Theorem 4.1) under the additionalassumption that the normal vector field of E ( r ) is constant outside a compactsubset of the ambient space. The proof of Theorem 3.1 strongly relies on geo-metric and economic properties: the classification of ruled minimal submanifoldsof the Euclidean space, the bundle structure of the equilibrium manifold and thepositiveness of prices. On the other hand, the proof of Theorem 4.1 combines deepgeometric results relating the topology of a minimal submanifold of the Euclideanspace with the fact that E ( r ) is globally diffeomorphic to an Euclidean space.It is worth noticing that the choice of a metric depends on the analysis. In[12] the metric on the equilibrium manifold was chosen to deal with asymptoticproperties related to economies with an arbitrarily large number of equilibria. In[13] the metric used was a tool to explore geometric properties which are intrinsic,i.e. they do not depend on the ambient space. But the purpose, and the approach,of the present work is entirely different and this affects the choice of the metricused.We believe that this information-theoretic and geometric approach can be fur-ther extended in different directions. Following the seminal contribution by [22](see also [4, 5] and [15]), an entropy-based metric could be developed in orderto compute geodesics representing redistributive policies. Another direction ofresearch (see [19] and [9]) is to analyze the extrinsic uncertainty in N ⊂ E ( r )caused by coalitional manipulation of endowments. This approach could providenew insights into the understanding of coalition formation and sunspot equilibria.Finally, due to the economic relevance of the shape of the equilibrium manifold, itcan be worth investigating the connection between its shape and the primitives ofthe model, an issue still largely unexplored. This local-global view can hopefullylead to new perspectives on issues like uniqueness and stability (see [10, 16] for asurvey).This paper is organized as follows. Section 2 recalls notations, definitions andthe existing results which will be used to prove our main results. Section 3 andSection 4 prove our main results, Theorem 3.1 and Theorem 4.1. The economic setup is represented by a pure exchange smooth economy with L goods and M consumers under the standard smooth assumptions (see [2, Chapter4]). The set of normalized prices is defined by S = { p = ( p , . . . p L ) ∈ R L | p l > , l = 1 , . . . , L, p L = 1 } and the set Ω = ( R L ) M denotes the space of endowments ω = ( ω , . . . , ω M ), ω i ∈ R L . The equilibrium manifold E is the set of the pairs ( p, ω ) ∈ S × Ω, whichsatisfy the equality: M X i =1 f i ( p, p · ω i ) = M X i =1 ω i , (2)where f i ( p, w i ) is consumer’s i demand.By [2, Lemma 3.2.1], E is a (closed) smooth submanifold of S × Ω, globally dif-feomorphic to S × R M × R ( L − M − = R LM , i.e. φ | E ∼ = R LM , where the smoothmapping φ : S × Ω → S × R M × R ( L − M − is defined by ( p, ω . . . , ω M ) ( p, p · ω , . . . , p · ω M , ¯ ω , . . . , ¯ ω M − ) , where ¯ ω i denotes the first L − ω i , for i = 1 , . . . , M − E : • the set of no-trade equilibria T = { ( p, ω ) ∈ E | f i ( p, p · ω i ) = ω i , i = 1 , . . . , M } ; • the fiber associated with ( p, w , . . . , w M ) ∈ S × R M , which is defined as theset of pairs ( p, ω ) ∈ S × Ω such that: – p · ω i = w i for i = 1 , . . . , M ; – P i ω i = P i f i ( p, w i ).By defining the two smooth maps f : S × R M → S × R LM , where f ( p, w , . . . , w M ) = ( p, f ( p, w ) , . . . , f M ( p, w M )), and φ F iber : E → S × R M , where φ F iber ( p, ω , . . . , ω M ) = ( p, p · ω , . . . , p · ω M ), since f ( S × R M ) = T ⊂ E and φ F iber ◦ f is the identity mapping, by applying [2, Lemma 3.2.1], Balasko shows [2,Proposition 3.3.2] that T is a smooth submanifold of E diffeomorphic to S × R M .5y construction, every fiber associated with ( p, w , . . . , w M ) is a subset of E which is the inverse image of ( p, w , . . . , w M ) via the mapping φ F iber . It is intu-itively clear that while holding ( p, w , . . . , w M ) fixed and letting ω varying alongthe fiber, there are not any nonlinearities which may arise from the aggregate de-mand. In fact the fiber is a linear submanifold of E of dimension ( L − M − E can be thought as a disjoint union of fibers parametrizedby the no-trade equilibria T via the mapping φ | E : E → S × R M × R ( L − M − :for a fixed ( p, w , . . . , w M ) ∈ S × R M , each fiber is parametrized by ¯ ω , . . . , ¯ ω M − .By letting ( p, w , . . . , w M ) varying in S × R M , we obtain the bundle structure ofthe equilibrium manifold.If total resources are fixed, the equilibrium manifold is defined as E ( r ) = { ( p, ω ) ∈ S × Ω( r ) | M X i =1 f i ( p, p · ω i ) = r } , (3)where r ∈ R L is the vector that represents the total resources of the economy andΩ( r ) = { ω ∈ R LM | P Mi =1 ω i = r } .Let B ( r ) = { ( p, w , . . . , w M ) ∈ S × R M | M X i =1 f i ( p, w i ) = r } (4)be the set of price-income equilibria (see [2, Definition 5.1.1]). B ( r ) is a sub-manifold of S × R M diffeomorphic to R M − [2, Corollary 5.2.4] through the map θ : S × R M → R L × R M − , defined by( p, w ) ( X i f i ( p, w i ) , u ( f ( p, w ) , . . . , u M − ( f M − ( p, w M − )) . (5)The equilibrium manifold E ( r ) is a submanifold of S × Ω( r ) diffeomorphic to R L ( M − [2, Corollary 5.2.5] φ ( E ( r )) = B ( r ) × R ( L − M − . (6)Moreover we can define and T ( r ) = T ∩ S × Ω( r ). By construction, even in afixed total resource setting, the equilibrium manifold preserves its bundle structureproperty and, hence, E ( r ) can be written as the disjoint union E ( r ) = ⊔ x ∈ T ( r ) F x , (7)where F x is an ( L − M − R L ( M − .6et t = ( t , . . . , t l − ), ¯ ω j = ( ω , . . . , ω l − ) and p ( t ) = ( p ( t ) , . . . , p l − ( t ). Fol-lowing [2] and [13], we can parametrize B ( r ) via the map: ϕ : R M − → B ( r ) , t → ( p ( t ) , w ( t ) . . . , w m − ( t )) (8)and E ( r ) via the map: Φ : R L ( M − → E ( r ) , (9)( t, ω , . . . , ω M − , . . . , ω , . . . , ω L − M − ) ( p ( t ) , ¯ ω , w ( t ) − p ( t ) · ¯ ω , . . . , w M − ( t ) − p ( t ) · ¯ ω M − )We end this section with the following result due to Balasko, deeply related tothe issue raised in this paper. Theorem 2.1 [2, p. 188 Theorem 7.3.9 part (2)]
If for every ω ∈ Ω( r ) there isuniqueness of equilibrium, the equilibrium correspondence is constant: The equi-librium price vector p associated with ω does not depend on ω . Remark 2.2
Hence (global) uniqueness implies (MEP) for E ( r ) under the as-sumption of a uniform distribution. This theorem will be used to prove the “onlyif” part of Theorem 3.1 and Theorem 4.1. M = 2 In this section we prove the following:
Theorem 3.1
Let M = 2 and assume a uniform distribution. Then E ( r ) satisfiesthe (MEP) if and only if the price is unique. Before proving the theorem we need some definitions. • a submanifold M n ⊂ R n + k is said to be ruled if M n is foliated by affinesubspaces of dimension n − R n + k . • a generalized helicoid is the ruled submanifold M n ( a , . . . , a k , b ) ⊂ R n + k , k ≤ n , admitting the following parametrization:( s, t , . . . , t n − ) ( t cos( a s ) , t sin( a s ) , . . . , t k cos( a k s ) , t k sin( a k s ) , t k +1 , . . . , t n − , bs )) , where a j , j = 1 , . . . , k , and b are real numbers (we are not escluding that oneof these coefficients could vanish and the generalized helicoid becomes an affinesubspace).The key ingredient in the proof of Theorem 3.1 is the following classificationresult on ruled minimal submanifolds of the Euclidean space. We refer the readerto [8, Section 1] and references therein (in particular [14] for a proof).7 heorem 3.2 ([14]) A minimal ruled submanifold M n ⊂ R n + k is, up to rigidmotions of R n + k , a generalized helicoid. We need also the following simple but fundamental fact:
Lemma 3.3
Let M n ( a , . . . , a k , b ) ⊂ R n + k be a generalized helicoid such that b · Q ki =1 a i = 0 . Then M n intersects any affine hyperplane of R n + k . Proof:
In cartesian coordinates x , y , . . . , x k , y k , x k +1 , . . . , x n − , x n an hyperplaneof R n + k has equation: α x + β y + · · · + α k x k + β k y k + α k +1 x k +1 + · · · + α n − x n − + α n x n = δ, where α i , β i , i = 1 , . . . k , α j , j = k + 1 , . . . n and δ are real numbers such that k X i =1 ( α i + β i ) + n X j =1 α j = 0 . On the other hand the following equation represents the condition to be satis-fied for a point of the hyperplane to belong to the generalized helicoid: k X i =1 t i ( α i cos( a i s ) + β i sin( a i s )) + n − X j = k +1 α j t j + α n bs = δ. Since one can always find a pair ( s , t ) satisfying the previous equation, the lemmais proved. (cid:3) Proof of Theorem 3.1:
Since (MEP) is equivalent to (MVP), E ( r ) is a minimalsubmanifold of S × Ω( r ). Since M = 2, by the bundle structure property (see (7)above) E ( r ) is a ruled submanifold in R L − . By Theorem 3.2, E ( r ) is (up to rigidmotions) a generalized helicoid. If some a i or b are equal to zero then E ( r ) is anhyperplane and, by Theorem 2.1, the price is unique. Otherwise if b · Q i a i = 0,by combining Lemma 3.3 with the fact that E ( r ) is contained in the open set of R L − consisting of those points with p > p being the price) one deduces that E ( r ) is an affine hyperplane and so the price is unique. The “only if” part followsby Theorem 2.1 (see Remark 2.2). (cid:3) A rigid motion of the Euclidean space R l is an isometry of R l given by the composition ofan orthogonal l × l matrix and a translation by some vector v ∈ R l . emark 3.4 In the previous theorem we use the fact that E ( r ) ⊂ S × Ω( r )is a minimal submanifold We can prove the same result by only assuming thatthe no-trade equilibria T ( r ) (which is one dimensional for M = 2) is a minimalsubmanifold of E ( r ), namely it is a geodesic. Indeed, by using the diffeomorphismbetween T ( r ) and B ( r ), and the parametrization Φ of E ( r ) (see (9)), T ( r ) can beparametrized through Φ by letting v = 0:Φ( t,
0) = γ ( t ) . Hence, if T ( r ) is a geodesic in E ( r ), its acceleration γ ′′ ( t ) is parallel, for every t ,to the unit normal vector N ( t ) | v =0 of E ( r ) or, equivalently, γ ′′ ( t ) ∧ N ( t ) | v =0 = 0.We have γ ′′ ( t ) = β ′′ ( t ) = (¨ p, , ¨ w ) and, since v = 0, Φ t ∧ Φ v = ˙ β ∧ δ = ( − ˙ w, p ˙ p, ˙ p ).Hence γ ′′ ( t ) ∧ N ( t ) | v =0 = β ′′ ∧ ( β ′ ∧ δ ) = 0 if and only if( − p ˙ p ¨ w, p ¨ p + ˙ w ¨ w, p ˙ p ¨ p ) = (0 , , . This implies that, for every t , ˙ p ¨ p = 0, i.e. ( ˙ p ˙ p ) ′ = 0, hence p is (constant and)unique. L = 2 In this section we consider an economy with two goods and an arbitrary numberof consumers. In this case the equilibrium manifold is a hypersurface. Moreprecisely, the equilibrium manifold E ( r ) has dimension R M − and the ambientspace has dimension R M − . So it makes sense to consider the normal vector field N along E ( r ), namely for each x ∈ E ( r ) we consider a unit vector N ( x ) parallelto the affine line T x X ⊥ normal to the tangent space T x X of X at x . The smoothmap N : E ( r ) → S M − which takes x to the point N ( x ) of the unit sphere S M − ⊂ R M − is called the Gauss map . Obviously, if the Gauss map is constantthen the price is constant and hence E ( r ) is an affine hyperplane in R M − . Inthe following theorem, which represents the second main result of the paper, weshow that the minimality assumption together with the constancy of the Gaussmap outside a compact set imply uniqueness of the equilibrium price. Theorem 4.1
Let L = 2 . Assume that the Gauss map is constant outside a com-pact subset of E ( r ) . Under the assumption of uniform distribution, E ( r ) satisfies(MEP) if and only if the price is unique. This theorem can be intepreted by saying that if the equilibrium manifold isminimal and there exists a compact subset K of R M − such that ( R M − \ K ) ∩ E ( r ) is the union of open subsets of hyperplanes each parallel to the hyperplane9 = const , then E ( r ) is indeed an hyperplane. As a consequence, the usual one-dimension representation of the equilibrium manifold cannot be minimal (see figurebelow). N ( y ) y N ( x ) x NE ( r ) K S M − The proof of Theorem 4.1 relies on the following theorem obtained in turn bysuitably combining some deep results obtained by Anderson in [1].
Theorem 4.2
Let M n ⊂ R n +1 , n > , be a minimal hypersurface such that thefollowing conditions are satisfied:1. M n has one end;2. M n is a C -diffeomorphic to a compact manifold ¯ M n punctured at a finitenumber of points { p i } .3. the Gauss map N : M n → S n extends to a C -map of ¯ M n .Then M n is an affine n -plane. Remark 4.3
The number of ends of a smooth manifold is a topological invariantwhich, roughly speaking, measures the number of connected components “at in-finity”. The reader is referred to [1] for a rigorous definition. What we are goingto use in the proof of Theorem 4.1 is that for n >
1, the Euclidean space R n hasonly one end. This is because R n \ K has only one unbounded component for anycompact set K . Proof of Theorem 4.1:
Since (MEP) is equivalent to (MVP), E ( r ) is a minimalsubmanifold of S × Ω( r ). If M = 2 we can apply Theorem 3.1. We can thenassume M > E ( r ) = 2 M − >
2. Hence, in order to prove the “if”part it is enough to verify that the three conditions of Theorem 4.2 are satisfied for E ( r ) ⊂ R M − . Condition 1 follows by the previous remark, since E ( r ) is globallydiffeomorphic to R M − . Notice that the unit sphere S M − is the Alexandroffcompactification of E ( r ) ∼ = R M − , namely it can be obtained by adding onepoint, called ∞ , to E ( r ). In other words E ( r ) is diffeomorphic to the sphere10 M − punctured to ∞ and so also condition 2 holds true. The assumption thatthe Gauss map N : E ( r ) → S M − is constant outside a compact set K means that N ( x ) = N , where N is a fixed vector in S M − , for all x ∈ E ( r ) \ K . Therefore, onecan extend N to a C ∞ -map ˆ N : S M − → S M − by simply defining ˆ N ( ∞ ) = N ,and so also condition 3 is satisfied. The “only if” part follows by Theorem 2.1 (seeRemark 2.2). (cid:3) Remark 4.4
Given a submanifold M n of a Riemannian manifold N n + k , one canexpress the minimality condition in terms of the vanishing of the mean curvature H . If k = 1, namely when M n is an hypersurface, the minimality condition, namely H = 0 , is equivalent to the vanishing of the trace of the differential of the Gaussmap (see e.g. [3]). Thus, for L = 2 one could try to show that minimality of E ( r )implies uniqueness of the equiilbrium price without imposing the extra conditionon the constancy of the Gauss map outside a compact set (as in Theorem 4.1) bycomputing the Gauss map through the parametrization (9) above and imposingthat the vanishing of the trace of its differential. This gives rise to a complicatedPDE equation, which the authors were not able to handle even when M = 3. References ∼ anderson/compactif.pdf[2] Balasko, Y., 1988, Foundations of the Theory of General Equilibrium, AcademicPress, Boston.[3] Carmo, M. do, 1992, Riemannian Geometry, Birkh¨auser, Boston.[4] Cowell, F., 1980, Generalized entropy and the measurement of distributional change,European Economic Review, 13, 147-159.[5] Cowell, F., 2011, Measuring inequality, Oxford University Press.[6] Cover, M. T, Thomas, J. A., 2006, Elements of Information Theory, Wiley, NewJersey.[7] DeMichelis, S., Germano,F. , 2000, Some consequences of the unknottedness of theWalras correspondence, Journal of Mathematical Economics 47 , 537-545.[8] Dillen, F., 1992, Ruled submanifolds of finite type, Proc. of the American Mathe-matical Society, 114, 3, pp. 795-798.[9] Goenka, A., Matta, S., 2008, Manipulation of endowments and sunspot equilibria,Economic Theory, 36, 267-282.
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